Sharp reiteration theorems for the K-interpolation method in limiting cases are proved using two-sided estimates of the K-functional. As an application, sharp mapping properties of the Riesz potential are derived in a limiting case.
We compute the K-functional related to some couple of spaces as small or classical Lebesgue space or Lorentz-Marcinkiewicz spaces completing the results of [12]. This computation allows to determine the interpolation space in the sense of Peetre for such couple. It happens that the result is always a GΓ-space, since this last space covers many spaces. The motivations of such study are various, among them we wish to obtain a regularity estimate for the so called very weak solution of a linear equation in a domain Ω with data in the space of the integrable function with respect to the distance function to the boundary of Ω.
Explicit formulae for theK-functional for the general couple((A0,A1)Φ0,(A0,A1)Φ1), where(A0,A1)is a compatible couple of quasi-normed spaces, are proved. As a consequence, the corresponding reiteration theorems are derived.
We prove optimal embeddings of homogeneous Sobolev spaces built over function spaces in R n with K-monotone and rearrangement invariant norm into other rearrangement invariant function spaces. The investigation is based on pointwise and integral estimates of the rearrangement or the oscillation of the rearrangement of f in terms of the rearrangement of the derivatives of f .
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